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Eurasian Math. J., 2013, Volume 4, Number 2, Pages 10–48 (Mi emj121)  

This article is cited in 24 scientific papers (total in 24 papers)

Approximate differentiability of mappings of Carnot–Carathéodory spaces

S. G. Basalaeva, S. K. Vodopyanovb

a Novosibirsk State University
b Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences

Abstract: We study the approximate differentiability of measurable mappings of Carnot–Carathéodory spaces. We show that the approximate differentiability almost everywhere is equivalent to the approximate differentiability along the basic horizontal vector fields almost everywhere. As a geometric tool we prove the generalization of Rashevsky–Chow theorem for $C^1$-smooth vector fields. The main result of the paper extends theorems on approximate differentiability proved by Stepanoff (1923, 1925) and Whitney (1951) for Euclidean spaces and by Vodopyanov (2000) for Carnot groups.

Keywords and phrases: approximate differentiability, Carnot–Carathéodory space.

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MSC: 53C17, 58C25, 28A75
Received: 27.09.2010

Citation: S. G. Basalaev, S. K. Vodopyanov, “Approximate differentiability of mappings of Carnot–Carathéodory spaces”, Eurasian Math. J., 4:2 (2013), 10–48

Citation in format AMSBIB
\by S.~G.~Basalaev, S.~K.~Vodopyanov
\paper Approximate differentiability of mappings of Carnot--Carath\'eodory spaces
\jour Eurasian Math. J.
\yr 2013
\vol 4
\issue 2
\pages 10--48

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    This publication is cited in the following articles:
    1. K. V. Storozhuk, “The Carathéodory–Rashevsky–Chow theorem for the nonholonomic Lipschitz distributions”, Siberian Math. J., 54:6 (2013), 1098–1103  mathnet  crossref  mathscinet  isi
    2. Karmanova M., Vodopyanov S., “A Coarea Formula For Smooth Contact Mappings of Carnot-Carath,Odory Spaces”, Acta Appl. Math., 128:1 (2013), 67–111  crossref  mathscinet  zmath  isi  scopus
    3. M. B. Karmanova, “Fine properties of basis vector fields on Carnot–Carathéodory spaces under minimal assumptions on smoothness”, Siberian Math. J., 55:1 (2014), 87–99  mathnet  crossref  mathscinet  isi
    4. S. G. Basalaev, “The Poincaré inequality for $C^{1,\alpha}$-smooth vector fields”, Siberian Math. J., 55:2 (2014), 215–229  mathnet  crossref  mathscinet  isi
    5. M. B. Karmanova, “An area formula for Lipschitz mappings of Carnot–Carathéodory spaces”, Izv. Math., 78:3 (2014), 475–499  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Karmanova M.B., “Fine Properties of Basis Vector Fields on Carnot-Caratheodory Spaces Under Minimal Smoothness Assumptions”, Dokl. Math., 89:3 (2014), 324–327  crossref  mathscinet  zmath  isi  elib  scopus
    7. Selivanova S., “Metric Geometry of Nonregular Weighted Carnot-Caratheodory Spaces”, J. Dyn. Control Syst., 20:1 (2014), 123–148  crossref  mathscinet  zmath  isi  elib  scopus
    8. Karmanova M., Vodopyanov S., “On Local Approximation Theorem on Equiregular Carnot-Caratheodory Spaces”, Geometric Control Theory and Sub-Riemannian Geometry, Springer Indam Series, 4, eds. Stefani G., Boscain U., Gauthier J., Sarychev A., Sigalotti M., Springer Int Publishing Ag, 2014, 241–262  crossref  mathscinet  zmath  isi  scopus
    9. M. V. Tryamkin, “The morphism property of subelliptic equations on the roto-translation group”, Siberian Math. J., 56:5 (2015), 936–954  mathnet  crossref  crossref  isi  elib  elib
    10. M. B. Karmanova, “Graph surfaces over three-dimensional Lie groups with sub-Riemannian structure”, Siberian Math. J., 56:6 (2015), 1080–1092  mathnet  crossref  crossref  mathscinet  isi  elib
    11. Karmanova M.B., “Graph Surfaces Over Three-Dimensional Carnot-Carath,Odory Spaces”, Dokl. Math., 92:1 (2015), 439–442  crossref  mathscinet  zmath  isi  elib  scopus
    12. M. B. Karmanova, “Polynomial sub-Riemannian differentiability on Carnot groups”, Dokl. Math., 94:3 (2016), 663–666  crossref  mathscinet  zmath  isi  elib  scopus
    13. A. V. Arutyunov, A. V. Greshnov, “Theory of $(q_1,q_2)$-quasimetric spaces and coincidence points”, Dokl. Math., 94:1 (2016), 434–437  crossref  mathscinet  zmath  isi  elib  scopus
    14. M. B. Karmanova, “Graph surfaces of codimension two over three-dimensional Carnot-Caratheodory spaces”, Dokl. Math., 93:3 (2016), 322–325  crossref  mathscinet  zmath  isi  elib  elib  scopus
    15. A. V. Greshnov, “$(q_1,q_2)$-quasimetrics bi-Lipschitz equivalent to $1$-quasimetrics”, Siberian Adv. Math., 27:4 (2017), 253–262  mathnet  crossref  crossref  elib
    16. M. B. Karmanova, “On an analogue of parallel translation on Carnot–Caratheodory spaces”, Dokl. Math., 96:3 (2017), 549–552  crossref  isi  elib
    17. A. V. Arutyunov, A. V. Greshnov, “Coincidence points of multivalued mappings in $(q_1,q_2)$-quasimetric spaces”, Dokl. Math., 96:2 (2017), 438–441  crossref  isi  elib
    18. M. B. Karmanova, “Approximation of Hölder mappings on Carnot–Caratheodory spaces”, Dokl. Math., 95:3 (2017), 199–202  crossref  isi  elib  elib
    19. M. B. Karmanova, “The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups”, Siberian Math. J., 58:2 (2017), 232–254  mathnet  crossref  crossref  isi  elib  elib
    20. A. V. Arutyunov, A. V. Greshnov, “$(q_1,q_2)$-quasimetric spaces. Covering mappings and coincidence points”, Izv. Math., 82:2 (2018), 245–272  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. M. B. Karmanova, “Three-dimensional graph surfaces on five-dimensional Carnot–Carathéodory spaces”, Siberian Math. J., 59:4 (2018), 657–676  mathnet  crossref  crossref  isi  elib
    22. M. B. Karmanova, “Area of some Hölder surfaces on Carnot-Carathéodory spaces”, Dokl. Math., 97:1 (2018), 73–76  crossref  mathscinet  zmath  isi  scopus
    23. S. G. Basalaev, “The local approximation theorem in various coordinate systems”, Siberian Math. J., 59:5 (2018), 778–785  mathnet  crossref  crossref  isi
    24. M. B. Karmanova, “Polynomial sub-Riemannian differentiability on Carnot–Carathéodory spaces”, Siberian Math. J., 59:5 (2018), 860–869  mathnet  crossref  crossref  isi
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